Maximum Likelihood for Matrices with Rank Constraints

Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry (Bertini) to compute all critical points of their likelihood functions. We present an intriguing duality conjecture that seems topological in nature.